Question

tan A/1+Sec A - tan A/1- Sec A = ?​

Answer 1

$\huge \bold { \underline{ \underline{answer}}} : - \\ \implies \: \bf{cosec \: A} \\ \\ \red{ \bold{ \underline{step - by - step \: explanation}} } : - \\ \\ A.T.Q. \\ \\ \implies \: \bold{\frac{ \tan \: A }{1 + sec \: A} - \frac{tan \: A}{1 - sec \: A}} \\ \\ \bold{ Taking \: LCM} \\ \\ \implies \: \bf{\frac{tan \: A(1 - sec \: A) - tan \: A(1 + sec \: A)}{(1 + sec \: A)(1 - sec \: A)} } \\ \\ \implies \: \bf{ \frac{tan \: A - tan \: A \: sec \: A - tan \: A - tan \: A \: sec \: A}{ {1}^{2} - {sec}^{2} A}} \\ \\ \implies \: \bf{ \frac{ - 2tan \: A \: sec \: A}{ - {tan}^{2} A} } \\ \\ \implies \: \bf{ \frac{sec \: A}{tan \: A} } \\ \\ \bf{\implies \: sec \: A \: cot \: A} \\ \\ \implies \: \bf{\frac{1}{cos \: A} . \frac{cos \: A}{sin \: A} } \\ \\ \implies \: \bf{\frac{1}{sin \: A} } \\ OR, \\ \implies \boxed{\bf{ \: cosec \: A}} \\ \\ \red{ \bf{formula \: used}} \\ \\ \star \: \bf{ \frac{1}{sin \theta} = cosec \theta} \\ \bf{\star \: 1 - {sec}^{2} \theta = - {tan}^{2} \theta} \\ \star \bf{ \: cot \theta= \frac{cos \theta}{sin \theta} }$